Off-line Calibration of Dynamic Traffic Assignment Models by Ramachandran Balakrishna Bachelor of Technology in Civil Engineering Indian Institute of Technology, Madras, India (1999) Master of Science in Transportation Massachusetts Institute of Technology (2002) Submitted to the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Field of Transportation Systems at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2006 c Massachusetts Institute of Technology 2006. All rights reserved. Author .............................................................. Department of Civil and Environmental Engineering May 12, 2006 Certified by .......................................................... Moshe E. Ben-Akiva Edmund K. Turner Professor of Civil and Environmental Engineering Thesis Supervisor Certified by .......................................................... Haris N. Koutsopoulos Associate Professor of Civil and Environmental Engineering, Northeastern University Thesis Supervisor Accepted by ......................................................... Andrew Whittle Chairman, Departmental Committee for Graduate Students
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Off-line Calibration of Dynamic Traffic Assignment Models
by
Ramachandran Balakrishna
Bachelor of Technology in Civil EngineeringIndian Institute of Technology, Madras, India (1999)
Master of Science in TransportationMassachusetts Institute of Technology (2002)
Submitted to the Department of Civil and Environmental Engineeringin partial fulfillment of the requirements for the degree of
Doctor of Philosophy in the Field of Transportation Systems
Chairman, Departmental Committee for Graduate Students
2
Off-line Calibration of Dynamic Traffic Assignment Models
by
Ramachandran Balakrishna
Submitted to the Department of Civil and Environmental Engineeringon May 12, 2006, in partial fulfillment of the
requirements for the degree ofDoctor of Philosophy in the Field of Transportation Systems
Abstract
Advances in Intelligent Transportation Systems (ITS) have resulted in the deploymentof surveillance systems that automatically collect and store extensive network-widetraffic data. Dynamic Traffic Assignment (DTA) models have also been developed fora variety of dynamic traffic management applications. Such models are designed toestimate and predict the evolution of congestion through detailed models and algo-rithms that capture travel demand, network supply and their complex interactions.The availability of rich time-varying traffic data spanning multiple days thus providesthe opportunity to calibrate a DTA model’s many inputs and parameters, so that itsoutputs reflect field conditions.
The current state of the art of DTA model calibration is a sequential approach, inwhich supply model calibration (assuming known demand inputs) is followed by de-mand calibration with fixed supply parameters. In this thesis, we develop an off-lineDTA model calibration methodology for the simultaneous estimation of all demandand supply inputs and parameters, using sensor data. We adopt a minimization for-mulation that can use any general traffic data, and present approaches to solve thecomplex, non-linear, stochastic optimization problem. Case studies with DynaMIT,a DTA model with traffic estimation and prediction capabilities, are used to demon-strate and validate the proposed methodology. A synthetic traffic network with knowndemand parameters and simulated sensor data is used to illustrate the improvementover the sequential approach, the ability to accurately recover underlying model pa-rameters, and robustness in a variety of demand and supply situations. Archivedsensor data and a network from Los Angeles, CA are then used to demonstrate scal-ability. The benefit of the proposed methodology is validated through a real-timetest of the calibrated DynaMIT’s estimation and prediction accuracy, based on sen-sor data not used for calibration. Results indicate that the simultaneous approachsignificantly outperforms the sequential state of the art.
Thesis Supervisor: Moshe E. Ben-AkivaTitle: Edmund K. Turner Professor of Civil and Environmental Engineering
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Thesis Supervisor: Haris N. KoutsopoulosTitle: Associate Professor of Civil and Environmental Engineering, NortheasternUniversity
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Acknowledgments
This thesis would not have been possible without contributions from various quar-
ters. Foremost, I would like to acknowledge the support and inputs from my thesis
supervisors, Professors Moshe Ben-Akiva and Haris N. Koutsopoulos. They have set
extremely high standards for me, and have taught by example.
My doctoral committee has been an invaluable source of suggestions, advice and
encouragement. I would like to thank Prof. Nigel Wilson, Dr Kalidas Ashok and Dr
Tomer Toledo for their support and guidance.
Faculty and friends have contributed immensely through informal discussions out-
side the classroom/lab. Their genuine interest in my work has been a source of en-
couragement, and has helped me place this research in perspective. I thank Professors
Nigel Wilson, Patrick Jaillet, Cindy Barnhart, Joe Sussman, Ikki Kim, Michel Bier-
laire and Brian Park, and PhD students Hai Jiang and Yang Wen, for their insights.
Other friends including Dr Arvind Sankar, Prof. Lakshmi Iyer, Dr K. V. S. Vinay
and lab-mates Vikrant Vaze and Varun Ramanujam have routinely buttonholed me
on my latest results, which has helped me clarify concepts in my own mind.
I am grateful to Dr Henry Lieu and Raj Ghaman of the Federal Highway Ad-
ministration, whose funding supported much of this research. The data for the Los
Angeles analysis was provided by the California PATH program, and Gabriel Murillo
and Verej Janoyan of the LA Department of Transportation. The tireless Dr Scott
Smith of the Volpe Center was instrumental in getting the outputs from this thesis
out into the real world.
CEE has a long list of able administrators who have cheerfully and pro-actively
attended to many a potential issue before they arose. I would especially like to thank
Leanne Russell, Anthee Travers, Cynthia Stewart, Donna Hudson, Pat Dixon, Pat
Glidden, Ginny Siggia and Sara Goplin for their constant assistance that resulted in
a smooth run through grad school.
Lab-mates come and go, but the memories will live on forever. I shared an office
and many cherished moments with Dr Constantinos Antoniou, who continues to be
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my fountain of knowledge on a wide range of transportation and IT topics. Together,
we proved the sufficiency of plain, vanilla e-mail for high-volume, real-time communi-
Figure 1-2: Off-line and On-line Model Calibration
Typically, models and the data needed to calibrate them fall in the same category.
Route choice models are thus traditionally calibrated using disaggregate survey data.
Similarly, modern OD matrix estimation methods rely on aggregate link sensor count
observations2. Given the paucity of disaggregate (survey) datasets, however, the
2An exception to this is the use of disaggregate OD surveys to generate OD matrices. This
32
analyst will often be faced with the task of calibrating complicated DTA models using
aggregate data alone. Alternatively, models estimated using traditional methods may
have to be updated using recent sensor measurements. This thesis provides a rigorous
treatment of these problems, and demonstrates how the disaggregate and aggregate
models within a DTA system may be calibrated jointly using aggregate data.
The off-line calibration of a DTA model is summarized in Figure 1-3. We have
a DTA model with a list of unknown inputs and parameters (dynamic OD flows,
route choice model parameters, capacities, speed-density relationships, etc). We must
obtain estimates of these inputs and parameters by using the information contained in
available aggregate, time-dependent traffic measurements, so that the DTA model’s
outputs accurately mirror the collected data. This data includes, but is not limited
to, counts and speeds from standard pavement loop detectors3. Different sets of
parameters may be estimated to reflect any systematic variability in traffic patterns
identified from several days of observed data. A priori estimates for some or all of the
parameters, if available, may serve as starting values to be updated with the latest
data.
Automated data collection technologies afford the measurement and storage of
large amounts of traffic data. This data is expected to span many days, representing
the various factors (demand patterns, supply phenomena, incidents, weather condi-
tions and special events) characteristic of the region. In order to apply the DTA model
in the future, a database of model inputs and parameters must be calibrated for each
combination of factors observed in the data. The calibration task must therefore
begin with data analysis that reveals these combinations, and partitions the sensor
measurements accordingly.
Once calibrated, an important practical consideration is the maintenance of the
historical database as sensor data from future days become available. The histori-
cal estimates of model inputs and parameters must be updated with every new day
of measurements. In keeping with this requirement, we focus on the development
method has largely been replaced in recent times by the approach based on link counts.3A more detailed description of aggregate data sources is provided in Section 3.2.
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Off-line Calibration
Field Measurements
Updated Estimates of
Simulator Parameters and Inputs
Simulator A priori
estimates
Figure 1-3: Calibration Framework
of a systematic calibration methodology that estimates all demand and supply vari-
ables using one day of data at a time. Methods to periodically update the database
have been proposed elsewhere (see, for example, Ashok (1996) and Balakrishna et al.
(2005a)), and are discussed in detail in Section 3.3.
1.4 Problem definition
Let the period of interest each day be denoted by H. This period could include the
entire day, or a specific portion of the day (such as the AM or PM peak). Let H
be divided into H intervals of equal duration, represented by h = 1, 2, . . . , H. Let
G denote the directed graph of nodes and links (and their associated characteristics,
including records of major incidents and special events) corresponding to the physical
transportation network input to the DTA model4. Let x represent the set of dynamic
OD flows xh , h ∈ H prevalent on that day. Each nOD-sized vector5 xh represents the
OD flows departing from their origin nodes during interval h. Further, let β be the
4We refer here to a general DTA model, chosen to suit the application at hand.5nOD denotes the number of OD pairs on the network.
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set of time-specific model parameters βh comprised of route choice model parameters
and supply-side variables such as link/segment output capacities and speed-density
function parameters. We denote the total set of unknown parameters [x β] as θ. Let
the travel time inputs to the route choice model be TTrc.
The off-line DTA calibration problem can now be defined as the simultaneous
estimation of all the demand and supply variables in θ, and a consistent set of route
choice travel times, error covariances and OD prediction model parameters, using
time-dependent counts and speeds recorded by traffic loop detectors. Traffic data M
are assumed to be available over the H intervals in H, so that
M = M1,M2, . . . ,Mh, . . . ,MH
The vector Mh contains records of vehicle counts and speeds recorded during interval
h.
A priori estimates xa and βa, if available, can provide valuable structural infor-
mation that must be exploited by the calibration methodology. The non-zero cells in
xa, for example, indicate the OD pairs that contribute to network flows (and hence
must be included as optimization variables). Further, speed-density or speed-flow
equations fitted to actual sensor data may serve as a good starting solution to be
refined through systematic calibration. Information about the relative magnitudes of
the various OD flows and model parameters may also be useful in speeding up the
optimization through the use of appropriate lower and upper bounds.
The network G can vary from day to day. For example, a subset of links or
lanes in the network might become unavailable for a few days due to severe incidents,
weather conditions or scheduled maintenance activities. Such disruptions are treated
as exogenous inputs to the calibration process. Details of planned special events that
are expected to have a significant impact on the day’s travel and traffic patterns are
also included in G.
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1.5 Thesis organization
The remainder of this thesis is organized as follows. Chapter 2 presents a detailed
review of existing DTA model calibration approaches, and identifies the strengths and
limitations of recent work in this area. Chapter 3 briefly describes typical DTA cali-
bration variables and available sensor data, and outlines our formulation of the off-line
DTA model calibration problem. Critical characteristics of the problem are analyzed,
and an efficient calibration methodology is developed. Further, optimization algo-
rithms are identified for solving the DTA calibration problem. The methodology is
systematically tested in Chapter 4, using the DynaMIT DTA system, and results from
a case study with a real dataset are presented and discussed in Chapter 5. Finally,
we conclude with a synthesis of our major findings, contributions and directions for
imates the gradient of the objective function through finite differences. Critically,
the approach infers the components of the gradient vector from two function evalua-
tions, after perturbing all components of the parameter vector simultaneously. The
computational savings are thus significant when compared to traditional stochastic
46
approximation methods, though many replications may be required in order to obtain
a more stable gradient through smoothing (Spall, 1994b). The Box-Complex algo-
rithm (Box, 1965) was applied with better success, though the number of convergence
iterations was still too few to study computational performance and the quality of
the final solutions.
2.3.2 Microscopic supply calibration
Although microscopic traffic models are not within the scope of this thesis, we review
a segment of literature on the calibration of such models. Some of the methods and
algorithms employed in this context are relevant to the problem at hand, and may be
appropriate for DTA model calibration after enhancement and modification.
The calibration of microscopic traffic simulation models has received serious atten-
tion in recent years, fueled by the widespread use of such models in professional and
academic circles. Early studies often relied on manual adjustments and heuristics
to reduce the discrepancy between observed and simulated quantities (see, for ex-
ample, Daigle et al. (1998), Gabriel Gomes and Adolf May and Roberto Horowitz
(2004) and Liu et al. (2004)). The time-consuming nature of this process, coupled
with lack of a systematic approach capable of handling large parameter sets, have
motivated research into the use of optimization algorithms to solve the calibration
problem.
Kurian (2000) describes an early attempt to use sophisticated optimization pack-
ages for the calibration of MITSIMLab, using data from 16 sensor stations in the
I-880 freeway corridor. He selects, through an experimental design, four parameters
that control deceleration characteristics in the car-following model (the time-varying
OD flows are obtained from a previous study, and are unchanged during calibration).
In his approach, the BOSS Quattro package is used to optimize the chosen parameters
using MITSIMLab as a black-box function evaluator. The algorithm, based on the
steepest descent concept, progresses by moving a certain step size along a direction
derived from the gradient at the current location. However, the inherent stochas-
ticity in MITSIMLab, together with the optimizer’s reliance on numerical gradients,
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resulted in noisy derivatives that prevented stable convergence. Further, the highly
nonlinear objective function resulted in the determination of different local optima
based on the starting parameter values selected.
Subsequent experiences with MITSIMLab involved the use of the Box-Complex
(Box, 1965) algorithm, a population-based approach that maintains a complex (or
set) of parameter vectors (points) and their corresponding objective function values.
The size of the complex was pre-determined based on the recommendations in Box
(1965). The algorithm begins by initializing the complex with points generated at
random, so as to cover the feasible region defined through lower and upper bounds
on each individual parameter. At every iteration, a point with the “worst” objective
function value is replaced by its reflection about the centroid of the remainder of
the complex, thus driving the complex towards the optimal solution. Darda (2002)
applies the Box-Complex method to estimate select car-following and lane-changing
model parameters under the assumption of a fixed OD demand matrix. However,
the convergence criterion on the maximum number of iterations was insufficient to
ascertain convergence.
The gradient-free downhill simplex algorithm (adapted from the Nelder-Mead sim-
plex procedure) was used by Brockfeld et al. (2005) to calibrate a small set of supply
parameters in a wide range of microscopic and macroscopic traffic models. Others
report on the successful application of genetic algorithms (GA) for the calibration
of select parameters in various microscopic traffic simulation tools (Abdulhai et al.,
1999; Lee et al., 2001; Kim, 2002; Kim and Rilett, 2003). The use of GA in trans-
portation is illustrated by Kim and Rilett (2004), who describe the calibration of
driving behavior parameters in the CORSIM and TRANSIMS microscopic models.
The data for the research consisted of traffic volume data from the I-10 and US-290
freeway corridors around Houston, TX. Apart from the simple structure (and the
corresponding lack of route choice) inherent to the test networks, the broader appli-
cability of their work is also limited by the small number of parameters estimated.
Indeed, the paper reports computational constraints even on such small examples!
The numerical values of several algorithmic constants were also assumed from prior
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analyses without sufficient elaboration. However, the authors do state the impact of
the OD matrix on the final results, though they do not include OD flows as variables
in the optimization.
Henderson and Fu (2004) provide a concise review of the transportation appli-
cations of GA to date. Indeed, the range of studies reported therein share several
common characteristics that limit the scope of their conclusions:
• All applications are in the domain of traffic micro-simulation, and focus en-
tirely on a subset of car-following and lane-changing parameters. Apart from
being few in number (the biggest problem instance involved 19 parameters),
OD flows were treated as exogenous to the GA application. This is a critical
limitation, as the estimation of OD flows will significantly increase the scale of
the optimization problem.
• The studies rarely compare the performance of GA against other well-established
non-linear optimization methods. Often, the primary measure of performance
is the improvement in the objective function value over the starting point, as
employed by Yu et al. (2005). The claimed superiority of GA is thus not clearly
established.
• GA involves a large set of highly sensitive tuning parameters and strategies
such as variable encoding schemes and crossover and mutation probabilities.
Most existing studies use default settings from earlier approaches, without any
analysis or justification for their use.
Systematic optimization techniques are thus being increasingly applied for the
calibration of supply model parameters. However, these experiences have been limited
to simple networks and small parameter sets. While some of the algorithms have
shown promise, tests on larger networks and variable sets should be performed to
ascertain their suitability for overall DTA model calibration.
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2.4 Estimation of demand models
Demand calibration involves the estimation of (a) travel behavior model parameters,
(b) time-varying flows for each OD pair on the network, and (c) other parameters
that the DTA model might use in the estimation and prediction of OD flows. We
begin the review of demand model estimation techniques with a discussion of travel
behavior model estimation.
The literature provides a rich spectrum of mode, departure time and route choice
models that capture driver behavior at both pre-trip and en-route levels. Pre-trip
decisions could include the choice of travel mode, departure time and route based
on perceptions of expected traffic conditions for the trip under consideration. Trip
chaining decisions (making one (or more) stop(s) before the final destination) and
multi-modal route selections (including park-and-ride transit options) may also be
made at the pre-trip stage. En-route decisions are made by drivers in response to
evolving trip conditions. The most common example of an en-route choice is to change
route due to unexpected traffic congestion, or in response to traveler information
obtained through a Variable Message Sign (VMS) or an on-board device (such as a
cell phone or radio).
The class of traveler behavior models considered here are disaggregate, in that
they predict the choices made by individual drivers (trip makers). In Section 2.4.1,
we briefly outline the standard discrete choice approach to estimating such models
using disaggregate survey data.
Off-line demand calibration at the aggregate level has primarily focused on the
estimation of OD flows from archived field measurements such as surveys, manual
traffic counts or automated loop detector counts. The OD estimation problem has
attracted substantial interest in the last few decades, and represents the calibration
of demand parameters (OD flows) that form critical inputs to any DTA system.
While OD estimation research covers both on-line (real-time) and off-line methods,
our review of the relevant literature (Section 2.4.2) will be limited to the off-line case.
Section 2.4.3 reviews work on the joint calibration of a DTA model’s demand
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models in an off-line setting. Such approaches capture the role of travel behavior
on the OD estimation problem, and attempt to incorporate their inter-relationships
while obtaining consistent estimates of the various OD flows and travel behavior
model parameters.
2.4.1 Travel behavior modeling
Discrete choice theory forms the backbone of most travel behavior analyses, and is
best illustrated in the context of DTA through route choice. The route choice model
contains the following dimensions:
• An individual driver n ∈ 1, 2, . . . ,N chooses from a set of alternatives (routes)
Cn.
• Driver n is described by a vector of characteristics. Each route i in the choice
set is similarly described by a vector of attributes. The combination of the
characteristics for driver n, along with the corresponding attributes for route i,
is represented by the vector Xin.
• Each driver n is assumed to perceive a “utility” associated with every route i
in his/her choice set. The utilities map the attributes and characteristics into
a real number for comparison.
• A decision rule is employed to determine the chosen route for each driver.
The principle underlying discrete choice theory is that of utility maximization:
each individual n will pick the route with the maximum perceived utility Ujn, j ∈ Cn.
From a modeling perspective, however, the utilities Ujn are not directly observed.
This discrepancy between the “true” utilities and their systematic model equivalents
is captured through Random Utility Theory:
Uin = Vin + εin (2.1)
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where Vin is a “systematic” utility computed as a linear function of the variables in
Xin. For example, Vin = β′Xin, where β is a vector of coefficients to be estimated.
The term εin represents the error between the model and the true utilities. Utility
maximization then yields the probability of driver n selecting route i as:
P(i) = Pr(Uin ≥ Ujn ∀j ∈ Cn) (2.2)
Combining Equations (2.1) and (2.2), we get:
P(i) = Pr(εin − εjn ≥ Vjn − Vin ∀j ∈ Cn) (2.3)
Assumptions on the distribution of the error terms εin (or the difference εin−εjn)
dictate the structure, richness and complexity of the resulting model. The assumption
of normally distributed errors, for example, results in the Multinomial Probit (MNP)
model, while Gumbel errors yield the popular Multinomial Logit (MNL) model.
The Probit model can potentially capture complex correlations among the alter-
native paths. However, its use involves the evaluation of high-dimension integrals that
do not possess closed-form solutions. The Logit model, with its attractive closed-form
expression, has thus been the most popular approach to capturing individual drivers’
travel decisions. Several variants and extensions to the above model classes have
been postulated, analyzed and tested, in view of the Logit model’s inability to handle
perceived correlations arising from the physical overlapping of alternative paths (the
well-known IIA property). These include the C-Logit and Path-Size Logit models,
and the flexible Logit Kernel approach. An overview of the various route choice model
structures is presented in Ramming (2001).
Traditional route choice model estimation (or the calibration of the vector β)
requires data from an individual (disaggregate) route choice survey. Each sampled
driver n responds with his/her characteristics (including both socio-economic vari-
ables as well as descriptors such as trip purpose), a set of alternative routes in his/her
choice set Cn, perceived route attributes and the chosen route.
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The vector β of unknown systematic utility coefficients is estimated using standard
concepts from Maximum Likelihood theory, by maximizing the joint probability of
the chosen paths in the dataset. A detailed mathematical treatment of the mechanics
of maximum likelihood estimation can be found in Ben-Akiva and Lerman (1985).
Disaggregate route choice models estimated from survey data possess several ad-
vantages. They provide a way of incorporating individual-specific characteristics and
tastes into the systematic utilities. The resulting estimates also largely reflect ac-
tually observed choices made by individual drivers. Sampling issues, however, im-
pose limitations on the choice-based model. A restricted sample size due to re-
source constraints and non-response may introduce bias in the estimated parameters,
as the resulting datasets may not be representative of the general driver popula-
tion. Justification bias (or a respondent’s tendency to provide data that validates
his/her choice) may further skew the resulting parameter estimates. Moreover, the
high costs associated with administering surveys introduces significant lag times that
could date the parameter estimates obtained. The use of aggregate data for enhanc-
ing route choice model estimation has only just begun to receive attention (exam-
ples include Ashok (1996), Tsavachidis (2000), Toledo et al. (2004), Jha et al. (2004)
and Balakrishna et al. (2005a)).
2.4.2 The OD estimation problem
The OD estimation problem has received significant attention in the fields of trans-
portation, computer network routing and general estimation theory. The problem
focuses on the inference of the elements of an unobserved matrix x of point-to-point
network trip demand3, based on aggregate traffic flow measurements y collected at
specific links on the network. The matrix x would have as many rows as there are
potential trip origin nodes, and as many columns as there are destinations. Each cell
in x thus represents the number of trips between a specific origin-destination pair.
Much of the OD estimation literature concentrates on the static problem. A single
3In the context of computer networks, demand may be defined in terms of data packets ratherthan vehicle trips.
53
OD demand matrix is estimated across a relatively large time period, such as an en-
tire day or the morning peak (see, for example, Cascetta and Nguyen (1988)). Static
approaches work with average flows across the entire study period. A critical limita-
tion of such methods is their inability to capture within-period temporal patterns in
OD demand, such as peaking. The demand inputs to DTA models must be dynamic,
to facilitate the modeling of time-dependent phenomena such as the formation and
dissipation of queues and spillback.
The following sections summarize the different approaches proposed for the esti-
mation of dynamic OD demand from observed sensor count measurements4. Common
to all methods is the assumption that the period of interest H is divided into intervals
h = 1, 2, . . . , H of equal duration. Let xh represent the matrix of OD flows departing
their origins during interval h, and yh the vehicle counts observed on various network
links at the end of h. The objective of the dynamic OD estimation problem is to
estimate the flows xh that replicate observed counts yh, ∀h ∈ H.
Least squares approach
The most widely employed dynamic OD estimation technique is based on extensions
to the least squares technique proposed by Cascetta and Nguyen (1988) in the static
context. Cascetta et al. (1993) propose a generalized least squares (GLS) framework
that fuses data from two sources to efficiently estimate dynamic OD flows. The
authors present two alternative estimators that work within this framework. The
sequential estimator optimizes for the unknown OD flows one interval at a time:
xh = arg minxh
[f1(xh,xah) + f2(yh, yh)] (2.4)
where xh is the current best solution; xah are a priori flows (extracted from other stud-
ies, or set to xh−1); yh are the fitted counts obtained by assigning xh to the network;
f1(•) and f2(•) are functions that measure the “distance” between the estimated or
fitted quantities from their a priori or observed values. It is generally expected that
4Time-varying vehicle counts are currently the most common source of field traffic data.
54
the number of link count observations (the dimension of yh) is much smaller than
the number of unknowns (the number of non-zero cells in xh). The a priori flows xah
thus provide valuable structural information that renders feasible a problem that is
otherwise indeterminate.
A measurement equation maps the OD flows xh to the counts yh through a linear
assignment matrix mapping:
yh =
h∑
p=h−p ′
aphxp + vh (2.5)
where the elements of aph specify the fractions of each OD flow in xp (departing
during interval p) that arrive at every sensor location during interval h. vh is an
error term. p ′ indicates the number of intervals spanning the longest trip on the
network, and is a function of network topology as well as congestion levels. Since the
sequential estimator constrains the flows in prior intervals to their best estimates, the
measurement equation may be re-written as:
yh = yh −
h−1∑
p=h−p ′
aphxp = ahhxh + vh (2.6)
Consistent with the GLS formulation, Equations 2.4 and 2.6 yield the following esti-
mator:
xh = arg minxh
[
(xh − xah)′W−1
h (xh − xah) + (yh − ahhxh)′R−1
h (yh − ahhxh)]
(2.7)
The above optimization is constrained so that xh ≥ 0. Wh and Rh are error variance-
covariance matrices that may be used to reflect the reliability of the different mea-
surements. Cascetta et al. propose setting them to identity matrices of appropriate
dimensions, in the absence of reliable estimates for the same.
The authors propose a second estimator that solves for the OD flows in multiple
55
intervals simultaneously:
(x1, x2, . . . , xH) = arg minx1,x2,...,xH
[f1(x1, x2, . . . ,xH;xa1 ,xa2 , . . . ,x
aH) (2.8)
+ f2(y1,y2, . . . ,yH; y1, y2, . . . , yH]
with Equation 2.5 serving as the measurement equation for counts.
A qualitative comparison of the two estimators is in order. The sequential ap-
proach estimates OD flows xh based only on yh, the first set of counts it contributes
to (these flows are fixed while estimating OD flows for subsequent intervals). Fu-
ture count measurements are thus not used to refine past estimates. The simulta-
neous estimator is more efficient in this regard, since it captures the contribution of
xh to the counts measured in all subsequent intervals. However, the approach in-
volves the calculation, storage and inversion of a large augmented assignment matrix,
which has been found to be too computationally intensive on large networks (see
Cascetta and Russo (1997), Toledo et al. (2003) and Bierlaire and Crittin (2004)).
The sequential approximation is therefore an attractive option for many applications.
Assignment matrices themselves are linear approximations of the relationship be-
tween OD flows and sensor counts. They are typically obtained through a network
loading model that mimics the progression of candidate OD flows along a set of paths
between each OD pair. Knowledge of drivers’ route choice behavior and network
travel times are thus essential to the computation of the assignment matrix, due to
their roles in splitting OD flows into path flows that can subsequently be propagated
along each path. Travel times, in turn, depend on the OD flows, whose “true” values
are as yet unknown. OD estimation in the presence of congestion is therefore a fixed-
point problem requiring an iterative solution methodology that captures the complex
dependencies between the OD flows, the route choice model and the network loading
model.
Cascetta and Postorino (2001) apply iterative schemes based on the method of
successive averages (MSA) to solve the fixed-point OD estimation problem and obtain
consistent OD flows and assignment matrices on congested networks. A GLS estima-
56
tor is used to generate updated flows in each iteration, which are then “smoothed” by
the MSA technique. The authors provide empirical evidence in support of a modified
OD smoothing algorithm (which they term MSADR5), that provides faster conver-
gence by re-initializing MSA’s iteration counter as the algorithm progresses. The
re-setting of the counter is performed with decreasing frequency. Intuitive arguments
are provided to show the equivalence of the final MSA and MSADR solutions. How-
ever, the approach pertains to the static case.
Kalman Filter approach
Ashok (1996) develops a sequential off-line OD smoothing scheme based on state-space
modeling concepts. This approach uses transition equations to capture the evolution
of system state, and measurement equations to incorporate the sensor count measure-
ments. The authors provide a key innovation over previous state-space approaches,
by defining the state in terms of deviations: the difference of OD flows xh from their
historical or expected values xHh . The above transformation allows the state to be
represented through symmetrical distributions (such as normal) that possess desirable
estimation properties, which would not be appropriate for OD flows directly.
A transition equation based on an autoregressive process describes the interval-
to-interval evolution structure for network state:
xh+1 − xHh+1 =
h∑
p=h−q ′+1
fph+1(xp − xHp ) + wh+1 (2.9)
where fph+1 is a matrix relating spatial and temporal OD relationships between in-
tervals p and h + 1. The parameter q ′ is the degree of the autoregressive process,
representing the length of past history affecting the current interval.
The measurement equation is obtained by adapting Equation 2.5 to work with
deviations:
yh − yHh =
h∑
p=h−p ′
aph(xp − xHp ) + vh (2.10)
5MSADR stands for MSA with Decreasing Reinitializations.
57
with yHh =∑h
p=h−p ′ aphx
Hp serving as historical count estimates. Rh and Qh denote
the error covariance matrices for vh and wh respectively. The author provides a
Kalman Filter solution approach for estimating xh: In a forward pass, the flows xh
are estimated sequentially for h = 1, . . . , H (ignoring the contribution of xh to counts
yh+1,yh+2, . . . ,yH). Each OD matrix is then re-estimated (updated) while back-
tracking from h = H to h = 1. The information contained in yH,yH−1, . . . ,yh+1 is
thus completely used in identifying the flows for interval h.
This work also contains modeling enhancements for practical applications. First,
state augmentation is proposed as a way of improving the efficiency of the esti-
mated OD flows by exploiting the information about prior OD departure intervals
(xh−1,xh−2, . . .) contained in sensor measurements yh. In this approach, OD devia-
tions from a pre-defined number of past intervals are added to the state vector, and
are re-estimated periodically as future intervals are processed. State augmentation
may be perceived as a compromise between the computationally attractive sequential
estimator, and its more efficient simultaneous adaptation.
Further, the author briefly discusses methods to estimate the initial inputs re-
quired by the Kalman Filter algorithm: the historical OD flows xHh , error covariance
matrices Qh and Rh, and autoregressive matrices fph , which are an important part of
off-line demand model calibration.
Maximum Likelihood (ML) approach
Hazelton (2000) presents an OD estimation methodology that uses traffic counts and
a priori OD flow estimates in a maximum likelihood framework. A theoretical esti-
mator is developed by assuming a general distribution for the OD flows and sensor
counts. Indeed, they show that the GLS formulations by Cascetta et al. (1993) may
be obtained by selecting the normal distribution to model all error terms. While this
approach presents an elegant generalization of some prior results, its applicability
to the DTA calibration context is limited. First, the analysis focuses on static OD
estimation. The further assumption of uncongested network conditions, while allow-
ing the author to ignore temporal dynamics in route choice fractions, renders the
58
approach unrealistic in the DTA context. Lastly, the general framework still requires
the assumption of a realistic distribution if the method is to become operational.
van der Zijpp’s approach
A method for estimating OD flows on freeway networks is developed by van der Zijpp
(1996), in which the time interval boundaries are determined by analyzing space-
time trajectories. Assuming that vehicle speeds are known, the trajectories of the
first and last vehicles in each departure interval are calculated. Trajectories for all
other vehicles departing during the interval are determined based on first-in, first-out
(FIFO) rules. The set of trajectories is then used to estimate split fractions that
allocate sensor counts to OD flows from the current and previous intervals. The split
fractions are modeled by a truncated multivariate normal (TMVN) distribution, and
are updated at each step through a Bayesian formula.
The above approach has been packaged into the DelftOD software (van der Zijpp,
2002), which has been applied in many freeway situations (see, for example, Hegyi et al.
(2003), Ngoduy and Hoogendoorn (2003) and Park et al. (2005)). However, the lack
of a closed-form expression for the TMVN distribution poses practical difficulties
when determining its mean. Further, the calculation of complete vehicle trajectories
requires knowledge of speeds during the entire trip. An accurate predictor of future
speeds or travel times is thus essential for real-world applications.
A note on assignment matrices
In his thesis, Ashok (1996) outlines two ways of obtaining an assignment matrix for
OD estimation. The simpler approach involves the use of a traffic simulator, say the
DTA model being calibrated, to load the current best OD flows onto the network.
The required fractions in the assignment matrix may then be calculated through a
simple book-keeping of vehicle records at sensors. However, recent experiences with a
network from Los Angeles (Gupta, 2005; Balakrishna et al., 2006) have indicated that
simulated assignment matrices may be sub-optimal for OD estimation. An issue of
particular concern centers around the stability of the calculated assignment fractions.
59
In the absence of a good starting OD matrix, artificial bottlenecks may result due
to spatial and temporal OD patterns that are far from optimal, yielding incorrect
assignment fractions. The use of small starting flows to offset this problem typically
results in highly stochastic and small fractions, since few vehicles will be assigned to
each path.
An alternative approach for calculating the assignment fractions from link travel
times has been discussed by Ashok (1996). Under certain assumptions regarding
vehicles’ within-interval departure times6, and with knowledge of time-dependent link
traversal times, one may calculate crossing fractions that represent the percentage of
each path flow departing during interval p that reaches every sensor during interval
h. The assignment fraction for a given sensor and OD pair may be computed by
summing across all paths (for the OD pair) the product of crossing fractions (to the
sensor in question) and the corresponding path choice fractions (obtained through
the application of a route choice model). Such analytically calculated assignment
matrices possess many advantages. First, the link travel times obtained from a traffic
simulator are average values computed from several vehicles (across different OD
pairs). The resulting assignment fractions are therefore less stochastic than those
obtained directly from the simulator. Secondly, uncongested (free-flow) travel times
are generally known with a high degree of accuracy, from observed sensor speed
data. The assignment fractions (and estimated OD flows) for the intervals leading
up to the congested regime are therefore accurate, and may be expected to yield
accurate travel times even in subsequent intervals when coupled with sequential OD
estimation. Further, the contributions of current OD departures on future intervals
accurately account for congestion (through the travel times), minimizing the effects
of the starting OD matrix. Finally, the calculated fractions capture all possible paths,
including those that may be assigned few or no vehicles during the simulation. The
last two points, however, lead to assignment matrices that are not as sparse as their
simulated counterparts, and significantly increase the time taken to solve the OD
6Vehicle departure times within an interval are assumed to be uniformly spaced, with the firstand last vehicles departing at the beginning and end of the interval.
60
estimation problem. Gupta (2005) reports on the improved convergence and fit to
counts when the assignment matrices are calculated from travel times.
The role of the assignment matrix in OD estimation underscores the importance
of route choice in demand model calibration. Some prior studies have strived to
estimate accurate network travel times (used by the route choice model) that are
consistent with the estimated OD flows. However, few have focused on the param-
eters of the route choice model itself. Often, these parameters are assigned ad hoc
or convenient values that are then fixed for the remainder of the OD estimation pro-
cedure. Mahmassani et al. (2003), for example, describe the calibration of dynamic
OD flows for DYNASMART using traffic sensor data and with assumed route choice
model parameters. The authors cite the general lack of calibration data as the reason
for assuming known route choice splits. These splits are hypothesized as outputs of
some other procedure, and are hence exogenous to the functioning of the DTA system.
Ignoring the role of the route choice model can lead to biased and inconsistent
estimates of travel demand. We now look at some literature related to the joint
estimation of OD flows and the route choice model parameters.
2.4.3 Joint estimation of OD demand and travel behavior
models
The demand simulator of a DTA model relies on estimates of OD demand, route
choice model parameters and network travel times in order to accurately model the
network and its underlying demand patterns. Demand calibration therefore involves
solving a fixed-point problem that explicitly includes the route choice model param-
eters as variables. Initial research calibrating OD flows treated route choice as exter-
nal to the OD estimation problem, potentially leading to biased OD flow estimates.
Cascetta and Nguyen (1988), for example, assume an ad hoc travel time coefficient
in the route choice model while estimating OD flows using a GLS approach.
Ashok (1996), while demonstrating his Kalman Filter based OD estimation
methodology, estimates a route choice model using traffic counts data from the A10
61
beltway in Amsterdam. The Logit route choice model contained a single coefficient,
that of travel time, whose optimal value (resulting in the best fit to the observed
counts) was identified through a line search that was conducted independent of the
OD estimation process. The final estimated parameter was very low, corresponding
to an all-or-nothing assignment to the shortest path. It should be noted that the
beltway provides exactly two paths between each OD pair, with little or no overlap
between them. An all-or-nothing assignment would therefore be reasonable for this
network, where one of the two routes is often much longer than the other.
Balakrishna (2002) uses multiple days of sensor counts to jointly calibrate dy-
namic OD flows and a route choice model within the DynaMIT traffic estimation
and prediction system. The study focused on an urban network from Irvine, CA
consisting of both arterial and freeway links. A static OD matrix for the AM peak
(available from the Orange County MPO through a previous planning exercise) was
adjusted systematically using a sequential GLS estimator to obtain dynamic OD ma-
trices for the entire AM peak. A Path-Size Logit based route choice model (Ramming,
2001; Ben-Akiva and Bierlaire, 2003) with three parameters was estimated using an
approach similar to the one outlined by Ashok (1996).
The joint calibration of DynaMIT’s route choice model and OD estimation and
prediction model using three days of sensor count data was carried out iteratively (a
detailed presentation of the algorithm may be found in Balakrishna et al. (2005a)).
Other estimated parameters included the error covariance matrices and autoregressive
factors used by DynaMIT’s OD estimation and prediction module. The performance
of the calibrated DynaMIT system was validated using two independent days of data
not used during calibration.
A related effort (Sundaram, 2002) develops a simulation-based short-term trans-
portation planning framework that jointly estimates dynamic OD flows and network
equilibrium travel times. While the coefficients of the route choice model are not esti-
mated, a consistent set of OD flows and travel times are obtained (for the given route
choice model) by iterating between an OD estimation module and a day-to-day travel
time updating model. The basis for the travel time iterations is a time-smoothing
62
procedure:
TTrck = λ TTk−1 + (1 − λ) TTrc
k−1 (2.11)
where drivers’ perceived route choice travel times TTrck on day k are modeled as
a function of the perceived estimates TTrck−1 from the previous day, and the latest
experienced (simulated) quantities TTk−1. The parameter λ captures a learning rate
(a value between 0 and 1) whose magnitude would be affected, among other factors,
by drivers’ familiarity with the network and its traffic patterns, and the prevalence
of traveler information services.
Sundaram’s approach operates in two steps. Travel times are established for a
given set of dynamic OD demands. The resulting equilibrium travel time estimates
are used to re-calculate assignment matrices for OD estimation. Travel times may
then be computed again based on the new OD estimates if convergence has not been
The discussion in this chapter leads to a definition of standard procedures adopted
when DTA models are currently calibrated using aggregate sensor data. The state-
of-the-art, as defined here, will form the reference (henceforth known as the reference
case, or Ref) against which the calibration methodology developed in this thesis will
be compared:
• Demand and supply models are calibrated independently (sequentially), ignor-
ing the effect of their interactions. Supply parameters are estimated first, then
demand parameters are calibrated with fixed supply models.
• Supply calibration:
– Capacities are estimated from sensor data and network geometry (primar-
ily the number of lanes), based on the Highway Capacity Manual (HCM).
Capacities during incidents are approximated from the HCM, based on the
number of affected lanes, total number of lanes and incident severity.
63
– Speed-density functions (or volume-delay relationships) are identified lo-
cally (ignoring network effects) by fitting appropriate curves to sensor data.
Links are grouped according to physical network features (such as the num-
ber of lanes and the position relative to diverge and merge points, on- and
off-ramps), and the most representative function is assigned to each group.
• Demand calibration:
– OD flows and route choice model parameters are estimated iteratively (se-
quentially).
– Time-dependent OD flows are estimated sequentially7 using one of several
methods (GLS or the Kalman Filter, for example) that rely on a set of
assignment matrices. The assignment matrices may be simulated, or com-
puted analytically based on the latest known travel times and route choice
parameters.
– Route choice parameters are estimated through manual line or grid searches.
A limited number of parameters might be handled in this way.
2.6 Summary
A review of the literature indicates several shortcomings in the state-of-the-art of
DTA model calibration, primary being the sequential treatment of demand and sup-
ply parameters. Most prevalent practices rely on heuristics and manual parameter
adjustment approaches that are largely based on judgment. Applications of system-
atic optimization algorithms for model calibration have been few, and focus primarily
on DTA model components. Moreover, these studies typically estimate a small subset
of parameters deemed important in explaining observed data for specific networks and
datasets, and typically do not perform sufficient iterations to ensure a high degree of
accuracy.
7As discussed earlier in Section 2.4.2, the sequential approach, while computationally attractive,may not accurately capture long trips encountered on large or highly congested networks.
64
Limited experience with simulation optimization in the realm of transportation
indicate the promise of select algorithms (such as the Box-Complex algorithm), but
not others (such as stochastic approximation or gradient-based methods). There is
need to explore this topic in depth, and develop a robust calibration methodology
that can simultaneously estimate both demand and supply model parameters in a
simulation-based DTA system. Chapter 3 presents a rigorous treatment of the off-line
DTA calibration problem, analyzes its dimensions and characteristics, and proposes
a robust and systematic estimator for its solution.
1998b,a, 1999)) provides huge savings in per iteration cost, by approximating the
gradient using just two function evaluations (independent of the value of K):
g(θi) =z(θi + ci∆i) − z(θi − ci∆i)
2ci
∆−1i1
∆−1i2
...
∆−1iK
(3.14)
where ∆i is a K-dimensional perturbation vector consisting of component-wise pertur-
bations ∆ik. Since the numerator in Equation 3.14 is invariant for all k = 1, 2, . . . , K,
the computational effort in each iteration is fixed (independent of K). This represents
an obvious benefit for scalability, though numerical tests will be required in order to
determine if the number of iterations to convergence is also reasonable. Neverthe-
less, from a theoretical perspective, SPSA represents a promising solution algorithm
that must be further evaluated to determine its suitability to the off-line calibration
problem.
The following steps describe the SPSA approach in detail:
1. The process is initialized (i = 0) so that θi = θ0, a K-dimensional vector of a
priori values. The SPSA algorithm’s non-negative coefficients a,A,c,α and γ
90
are chosen according to the characteristics of the problem3.
2. The number of gradient replications (grad rep) for obtaining the average gra-
dient estimate at θi is selected4.
3. The iteration counter is incremented: i = i + 1. The step sizes ai and ci are
calculated as ai = a/(A + i)α and ci = c/iγ.
4. A K-dimensional vector ∆i of independent random perturbations is generated.
Each element ∆ik, k = 1, 2, . . . , K, is drawn from a probability distribution
that is symmetrically distributed about zero, and satisfies the conditions that
both |∆ik| and E|∆−1ik | are bounded above by constants. The literature indicates
the suitability and success of the Bernoulli distribution (∆ik = ±1 with equal
probability). Note that the inverse moment condition above precludes the use
of the uniform or normal distributions.
5. The objective function is evaluated at two points, on “either side” of θi. These
points correspond to θi+
= θi + ci ∆i and θi− = θi − ci ∆i. Lower and upper
bound constraints are both imposed on each point before function evaluation.
6. The K-dimensional gradient vector is approximated as
g(θi) =z(θi
+
) − z(θi−)
2ci
∆−1i1
∆−1i2
...
∆−1iK
(3.15)
The common numerator for all K components of the gradient vector distin-
guishes the SPSA approach from traditional FD methods.
7. Steps 4 to 6 are repeated grad rep times, using independent ∆i draws, and an
average gradient vector for θi is computed.
3Some guidelines for the selection of SPSA parameters are outlined in a later section.4Gradient smoothing is believed to provide more stable approximations when function evaluations
are noisy.
91
8. An updated solution point θi+1 is obtained through the application of Equa-
tion 3.11. The resulting point is again adjusted for bounds violations.
9. Step 3 is re-visited until convergence. Convergence is declared when θi and the
corresponding function value z(θi) stabilize across several iterations.
The convergence characteristics of SPSA depend on the choice of gain sequences
ai and ci, and the distribution of the perturbations ∆i. Spall (1999) states
that the two gain sequences must approach 0 at rates that are neither too high nor
too low, and that the objective function must be several times differentiable in the
neighborhood of θ0. If these properties hold, and if the perturbations are selected
according to the requirements stated in Step 4 above, then the author provides the
best-case convergence rate as i−1/3. The paper also claims the superior efficiency of
SPSA compared to FDSA, with a K-fold saving per iteration.
Spall (1998a) provides some intuition for the success of SPSA, by comparing
its performance with that of FDSA. When the objective function may be evaluated
with little or no noise, FDSA is expected to emulate the steepest descent approach.
Mathematical analysis has established this descent direction to be at right-angles to
the contour line at each point. SPSA, owing to a random search direction, does not
necessarily follow a true descent to the optimum. That is, the gradient approximations
may differ from the true gradients, so that the corresponding search directions deviate
from those of steepest descent. However, SPSA’s gradient approximation is almost
unbiased:
E[g(θi)] = g(θi) + bi (3.16)
where the bias bi = µci2
, the term µ being a constant of proportionality. As ci → 0
(for large i), the small bias bi vanishes. The path determined by SPSA is thus
expected to deviate only slightly from that of FDSA. The “errors” in the search path
will average out across several iterations, so that FDSA and SPSA converge to the
same solution in a comparable number of iterations (Figure 3-3).
When the function evaluations are noisy, neither FDSA nor SPSA will trace the
steepest descent directions with certainty. However, extending the previous analogy,
92
Figure 3-3: SPSA vs. FDSA [Spall (1998a)]
the SPSA directions on average remain close to the optimal directions.
The SPSA method outlined in this section immediately illustrates the potential
computational savings for large-scale optimization. Unlike traditional stochastic ap-
proximation approaches, the effort expended per SPSA iteration is exactly 2 func-
tional evaluations, and is independent of the number of parameters, K. The sequence
ci, if chosen carefully, can overcome the other limitation of FDSA, namely the unre-
liability of gradient approximations based on very small perturbations. Essentially, ci
must approach zero with i, but must do so at a “reasonable” rate. Rapid convergence
of ci may result in unreliable derivatives, while a slow rate may prevent SPSA from
approaching the true gradient near the optimum.
93
3.6.2 Pattern search methods
Pattern search methods are often listed as direct search methods, since they do not
require derivative calculations. Instead, they compare function values to determine
the “best” direction of movement from the current solution.
Hooke and Jeeves method
This algorithm (Hooke and Jeeves, 1961) starts from an initial solution θ0 and begins
by determining a “good” search direction for k = 1, the first component of θ0. This
is achieved by moving by a certain search step size on either side of θ01, and compar-
ing the function values at both points (while maintaining the remaining parameter
dimensions k = 2, 3, . . . , K constant). If an improved point is not obtained, the step
size is reduced, and the search is repeated until a local descent direction is identified
for k = 1.
An intermediate point is generated by moving along this direction (keeping all
other components fixed). By definition, the objective function at the intermediate
point is lower than the value computed at θ0. The search for k = 2 starts from this
new point, to generate a second intermediate point. The process is repeated until a
search direction has been identified for each of the K components in the parameter
vector. The iteration ends by collating all K search directions into a single direction,
and applying an update that yields θ1, the “solution” from the first iteration. Sub-
sequent iterations repeat similarly, until some pre-defined termination thresholds are
satisfied. Note that an improvement in objective function at each iteration is not
guaranteed by this method. However, the iterations are expected to move towards
a local optimum as i → ∞. More details about this class of methods may be found
in Kolda et al. (2003).
The independence from derivatives is conceptually a very attractive feature of
pattern searches, since stochastic DTA models are generally expected to result in un-
reliable gradient estimates. However, the personal experiences of the author indicate
that these methods perform poorly when applied to even small- and medium-sized
94
simulation optimization problems. The primary reason for failure is the focus on the
immediate (component-wise) vicinity of the starting point, which is often far from a
global optimum. A new pattern requires at least K + 1 function evaluations. When
combined with a large-scale and highly non-linear function, the algorithm quickly
restricts its search space to a very small radius around the current point (due to a
rapidly shrinking search step size), thus making very slow progress even towards the
nearest local optimum. Search methods possessing the capability to “jump” rapidly
in a promising direction before refining their search, will therefore be better suited to
the calibration context.
Nelder-Mead (Simplex) method
The Nelder-Mead method (Nelder and Mead, 1965) maintains a population of K + 1
points (called a simplex), and begins by computing the objective function at each
point. At every iteration, the worst point is replaced by its reflection about the
centroid of the remaining points in the simplex. The objective function is evaluated
at the new point, and the process continues until little improvement can be achieved
by eliminating the worst point. The simplex can shrink or expand depending on the
objective function value of the new point, and ideally shrinks to a unique solution at
convergence. The initial simplex may be set up by randomly generating K+ 1 points
that satisfy the lower and upper bounds.
Convergence results for the Nelder-Mead algorithm are scarce, and are largely lim-
ited to the one- and two-dimensional cases (see, for example, Lagarias et al. (1998)).
This, and other papers, outline the difficulties in developing rigorous convergence
analyses when K > 2, and provide empirical evidence in support of the assertion that
the Nelder-Mead method can terminate at a sub-optimal point in noisy situations.
Box-Complex method
The Box-Complex algorithm (Box, 1965) is an extension to the Nelder-Mead ap-
proach: they both begin with a set of randomly-selected feasible points that span
the search space. In each iteration, a candidate point (one with the highest objective
95
function value) is replaced by its reflection about the centroid of the remaining points.
If the resulting point is worse than the candidate, the point may be moved closer to
the centroid using some contraction scheme. The algorithms aim to move in a di-
rection that eventually collapses the population into a single solution. The primary
difference between the two methods lies in the definition of the size of the population:
while Simplex requires exactly K + 1 points in its set, Box-Complex requires a mini-
mum of K+2. The use of a larger set can potentially increase the speed and accuracy
of the search, and also guard against the possibilities of numerical instabilities with
the Nelder-Mead approach (see Box (1965) for details).
Below we present an outline of the algorithm, tailored to our specific problem
instance (a more elaborate treatment can be found in the original paper by Box):
1. A complex S of S > K+1 points is generated5. Each point θs is a complete pa-
rameter vector of dimension K. Let θsk denote the kth component of the point θs,
s = 1, 2, . . . , S. The first point θ1 (for s=1) is set to θ0 (determined externally
by the modeler), and is assumed to satisfy all the constraints. The remaining
S − 1 points are obtained one at a time, through the following procedure:
θsk = lk + rsk (uk − lk) (3.17)
where rsk is a uniformly distributed random number in (0,1). The complex of S
points thus spans the feasible space defined by the bound constraints.
2. The function value z(θs) is evaluated at each point s in the complex. A point
s′ with the “worst” (largest) objective function value is determined6, and is
replaced by its reflection about the centroid of the remainder of the complex:
θs′
k = θk + α[
θk − θs′
k
]
(3.18)
where θk = 1S−1
∑s∈S\θs′ θ
sk is the centroid for dimension k. If bound constraints
5The need for this constraint on the complex size is motivated in a later discussion.6It should be noted that there can be multiple points with the same worst objective function
value. One of these points is chosen at random for replacement.
96
are violated, the point is moved an infinitesimal distance δ within the violated
lower or upper limit. The value of α is chosen to be greater than unity7 which
allows the complex to expand (if necessary), enabling rapid progress if the initial
best solution is far from the optimum. Typically, α = 1.3.
3. If a new (reflected) point repeats as the worst point on consecutive trials, it is
moved one half the distance towards the centroid of the remaining points:
θs′
k =θs
′
k + θk
2, k = 1, 2, . . . , K (3.19)
Bound constraints are checked and adjusted as previously outlined. The se-
lection of α > 1 helps to compensate for such adjustments that shrink the
complex.
4. The algorithm terminates when the function values of all points in the complex
satisfy some pre-determined distance measure. For example, iterations might
be stopped when all function values are within a certain percentage of each
other across consecutive iterations.
As mentioned earlier, the size S of the complex must be at least K+ 2, where K is
the number of parameters being estimated. When S equals K+ 1, the complex could
potentially collapse into a subspace defined by the first binding constraint, preventing
the exploration of other constraints. The Box-Complex method is therefore preferred
over the Nelder-Mead method. Box (1965) suggests a practical setting of S = 2K.
The advantages of the Complex algorithm in the context of calibrating simulation-
based functions are many. The non-dependence on numerical gradients has already
been stressed as a key feature. The random starting complex used by the algorithm
has the potential to move quickly towards the optimum before refining its search. In
addition, the random initialization of the complex significantly increases the chance
of converging to the global optimum, even if this is located “far” from the initial point
θ0 (since a subset of the starting complex is highly likely to be spread far away from
7Such an approach is known as over-reflection.
97
the starting solution).
A potential drawback of the Complex method is related to its focus on the worst
point in the complex. While the algorithm repeatedly expends effort to improve the
points with the highest objective function value, an improvement to the “best” point
(with the lowest function value) is not guaranteed at every iteration. When combined
with the fact that multiple function evaluations may be required per iteration (in case
the worst point repeats), the algorithm tends to display extremely slow convergence
rates as the optimization proceeds. Further, model stochasticity may result in an
apparently worst point being eliminated from the complex, when it should have been
retained.
3.6.3 Random search methods
Random search methods adopt probabilistic mechanisms to randomly select up-
dated parameter vectors with the hope of improving towards an optimum. They
are gradient-free, yet are characterized by a large set of tuning parameters that must
be selected (often on a case-by-case basis). They are more suited to the context of
discrete variable optimization over small search spaces. In the realm of large-scale
continuous optimization, random search methods have displayed slow convergence (if
they at all move towards the optimum). Simulated annealing and genetic algorithms
are two common types of random searches, which we review here.
Simulated annealing
Simulated annealing (Metropolis et al., 1953; Corana et al., 1987) is the optimization
equivalent of the physical process of cooling. The method begins with a very high
temperature (chosen by the modeler), and attempts to reach the lowest possible tem-
perature (of zero) just as heated metal cools towards room temperature. When met-
als re-crystallize from a high-energy state, their molecules might attain intermediate
states of higher energy while they re-align themselves through an adiabatic (equi-
librium) process. The optimization method follows an analogous “learning” process,
98
assigning a decreasing (non-zero) probability of traveling uphill while maintaining a
generally downward trend in the objective function. The early iterations therefore
allow for random “jumps” to escape from local optima.
Specific details of the implementation of simulated annealing methods vary widely
in the literature. However, the need for the pre-selection of a large number of tuning
parameters implies that significant effort may be required in identifying their optimal
values for each application. These parameters include (a) the initial temperature, (b)
the distribution of the perturbation applied to randomly generate updates, (c) the
cooling schedule that determines the sequence of temperatures, and (d) the criteria for
lowering the temperature (typically tied to the number of function evaluations of ran-
domly perturbed parameter vectors allowed at each temperature setting). The work
by Corana et al. (1987) is widely cited in the literature, and their tuning parameters
are largely adopted in applications.
Simulated annealing has been found to be effective in combinatorial optimiza-
tion problems, and has been widely applied in the area of electronic circuit de-
sign (Kirkpatrick et al., 1983). The primary advantage of the method is the ability
to reach a global optimum, due to the high initial probability of visiting a much
better solution by chance. The experience with continuous variables and large prob-
lems, however, is not encouraging. Wah and Wang (1999) adapt the basic algorithm
(ascribed to Metropolis et al. (1953)), to the case of constrained optimization with
continuous variables. They present many heuristics tailored to a set of benchmark
problems. Goffe et al. (1994) demonstrate that the tuning parameters suggested by
Corana et al. (1987) result in convergence for a 2-variable test case, after 852,001
function evaluations! The authors further propose modified parameters that reduce
this to 3,789 evaluations, which while being a significant reduction, is still high for
a small example. They note that the parameters in Corana et al. (1987) are conser-
vative, but acknowledge that they would be better suited for highly non-linear and
larger cases (indicating the method’s lack of scalability).
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Genetic algorithms
Genetic algorithms (GA) (Holland, 1975; Goldberg, 1989) are classified as evolution-
ary search methods, and are based on the theory of natural selection. A population
of starting solutions (chromosomes or individuals) is generated at random, and their
fitness (objective function values) evaluated. Solutions that are fitter (i.e. more likely
to be closer to an optimum) are retained in the population with a higher probability,
while the inferior points are eventually discarded. The population of current feasible
solutions progresses through generations (iterations), with several operators acting
on the chromosomes to decide the population that survives to the next iteration.
The fittest individuals in the current population are selected for starting a new
generation. The chosen individuals are crossed over in pairs with some probability,
and the resulting gene string may be further mutated to increase the randomness
of the new population (to perturb the search in a hitherto unexplored direction).
An elitist strategy might be enforced in order to retain the best solution(s) without
change. Fitness is evaluated before repeating the process.
Genetic algorithms have been applied to solve small transportation optimization
problems, including the calibration of a subset of microscopic traffic simulation model
inputs (see, for example, Kim and Rilett (2004) and Henderson and Fu (2004)),
however, they appear to be inferior to other methods reviewed earlier, for large-scale
calibration. GA are naturally tailored for integer variables. In this regard, they
differ fundamentally from other optimization methods that perform better in the
continuous domain. This characteristic is primarily because variables’ feasible values
are coded as binary strings. Discretizing the feasible ranges of a large number of
continuous variables (like the capacities and speed-density parameters encountered
in DTA models) would thus result in numerous possibilities, depending on the step
size chosen for this purpose. Even OD flows, if treated as integers, would result in an
extremely large set of potential values due to their significantly wider bounds.
Secondly, the algorithm requires several parameters to be pre-defined, includ-
ing crossover and mutation probabilities, a selection method and an elitist strategy.
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The choice of these parameters has been found to be critical, as demonstrated by
Mardle and Pascoe (1999). The authors report on the high sensitivity of GA to al-
gorithmic implementations, and the large running times when compared to more
traditional methods. They conclude with the recommendation that GA (and ran-
dom search methods in general) be viewed as an option only when other algorithms
fail. Henderson and Fu (2004) draw attention to the heuristic nature of GA, with no
guarantee of convergence. Unless the algorithm’s parameters are carefully selected
for each specific application, solutions of poor quality may result due to premature
“convergence”.
Thirdly, Henderson and Fu (2004) review an application of GA for maximum
likelihood estimation (Liu and Mahmassani, 2000), in which the effect of population
size and the number of generations (iterations) is briefly explored. The study found
that the two quantities varied greatly depending on the search space and the nature
of the objective function, and that large populations are necessary if high levels of
accuracy are desired. This conclusion is crucial in terms of scalability, especially when
function evaluations are expensive.
3.6.4 Summary
A wide variety of simulation optimization algorithms exist in the literature. However,
few have been tested on even medium-sized instances of non-linear optimization prob-
lems. While several approaches possess theoretically attractive properties desirable
for the calibration of large-scale traffic simulation models, their performance (includ-
ing both accuracy and running time) must be evaluated empirically before a suitable
algorithm(s) can be identified. Based on the preceding analysis and some preliminary
numerical experiments, three algorithms were short-listed for detailed testing:
• Box-Complex (pattern search)
• SNOBFIT (response surface method)
• SPSA (stochastic approximation - path search)
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3.7 Solution of the off-line calibration problem
In this section, we discuss some issues related to the application of the three selected
algorithms. First, we suggest the combination of Box-Complex and SNOBFIT to
increase the efficiency of the optimization. We then outline some guidelines for the
selection of algorithm parameters in the various methods.
3.7.1 Combined Box-SNOBFIT algorithm
We propose a solution approach that combines the Box-Complex and SNOBFIT op-
timization algorithms in a way that exploits their respective advantages. A review of
the merits and potential drawbacks of the two algorithms suggests a natural integra-
tion scheme to obtain accelerated convergence and added estimation accuracy.
The Complex algorithm has the ability to cover the feasible space effectively,
and rapidly replace the high-objective function points with estimates closer to the
initial best point. However, the rate of convergence can drop significantly as it be-
comes harder to improve the worst point just through reflections about the centroid.
SNOBFIT efficiently utilizes the information from every function evaluation to sys-
tematically search for local and global minima. However, it might benefit from a
good set of starting points that reduce the need for costly quadratic approximations
around remote points.
We propose a two-step approach in which the Complex algorithm first shrinks a
randomly generated initial set of points to one that is more uniform in function values,
without expending the computational resources needed to converge to the optimum.
The result is expected to be a complex that is more representative of the various
local minima of the non-linear objective function. In the second step, SNOBFIT
uses the final complex as the starting set L to further refine the search through
local approximation. A determination of the transition point (when the optimization
switches from Box-Complex to SNOBFIT) may be made after reviewing the progress
of the first method. A logical option would be to switch when the best objective
function in the complex flattens out consistently across successive iterations.
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While the per-iteration effort for SPSA represents a K-fold saving over FDSA or
population-based methods such as Box-SNOBFIT, analyses of the total time to con-
vergence must take into account the number of iterations required by each algorithm
to satisfy the stopping criteria. While a determination of the true running-time sav-
ings will require extensive empirical work (see Chapter 4), it should be noted that
the significant difference in per-iteration cost could, in theory, allow SPSA to per-
form many more iterations in the same (or less) time, to enable it to quickly identify
solutions very close in quality to the final Box-SNOBFIT result.
3.7.2 Some practical algorithmic considerations
Both Box-SNOBFIT and SPSA use coefficients that determine the actual workings
and performance of the respective algorithms. The choice of these coefficients sig-
nificantly impacts their practical rates of convergence. In this section, we borrow
from the literature and experimental experience to provide a brief review of the more
critical algorithmic coefficients, along with guidelines for their initialization.
Selecting algorithmic coefficients for Box
Perhaps the most critical input to the Box algorithm is the size of the population (or
complex) of points it maintains at each iteration. Theoretical considerations require
that this be a minimum of K + 2 to ensure numerical stability. A less dense set
risks a collapse of the complex along one or more dimension(s), with little chance
of pulling away into better regions of the feasible domain. Based on experiments on
fairly small problems, Box (1965) recommends that the complex size be twice or
thrice the problem dimension K, which immediately raises the issue of scalability: the
number of function evaluations required in order to set up the optimization iterations
equals the size of the complex. It should be noted, however, that each subsequent
iteration improves a single point in the set (one with the worst objective function).
The corresponding computational effort is generally very small relative to the initial
expense, though the per-iteration burden will typically increase sharply after many
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iterations. This increase is attributed to the fact that it gets progressively harder to
find, through a simple reflection operation about the shrinking centroid, a “better”
replacement for the current worst point. In other words, the worst point under
consideration may have to be reflected multiple times before its function value falls
below that of the second-worst point. Indeed, this behavior is a key contributor to the
drastic slow-down in Box’s convergence rate after the complex has shrunk significantly
in the early iterations.
Selecting algorithmic coefficients for SNOBFIT
Like the Box algorithm, SNOBFIT is also population-based. Therefore, the identi-
fication of an appropriate population size is again an important practical aspect. A
hard constraint in this regard is the requirement that there be enough points in the
population for SNOBFIT to perform its local quadratic fitting around each point.
Since this step involves the five nearest points for each point under consideration, a
minimum population size of K + 6 is necessary (Huyer and Neumaier, 2004).
As in any population-based method, maintaining a larger set of points enhances
SNOBFIT’s ability to locate a good solution. However, unlike the Box method,
increasing the size of the population adds to the computational time in every iteration.
This is because each SNOBFIT iteration recommends a new set of points (based on its
quadratic minimizations) at which the function must be evaluated before progressing
to the next iteration. From a scalability perspective, therefore, the minimum size of
K + 6 is the most attractive setting.
Another key input to SNOBFIT is p, the probability of recommending a type 3
point8. A smaller p encourages SNOBFIT to explore hitherto unknown areas of the
search space, in the search for a global solution. Larger values of p limit the search to
the vicinity of the points at which the objective function is already known. p = 0.3
was found to work well for the calibration problem.
8Type 3 points are chosen from the list of local minima obtained through quadratic fitting aroundeach point in the population. The best local minimum is designated as type 1, and is removed fromcontention for a type 3 label.
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Selecting algorithmic coefficients for SPSA
SPSA is a random-search method, and does not maintain a population of points. This
is perhaps the primary reason for its limited global optimization ability. However,
the algorithm requires the initialization of other key constants: a, A, c and α and γ.
Experiments have confirmed that the values of α = 0.602 and γ = 0.101 reported
in the literature are adequate in the context of the calibration problem. Spall (1998b)
also provides similar values associated with asymptotic optimality. The value of the
“stability constant” A = 50 was also found to work well. The choice of the remaining
two terms, however, can significantly impact the performance of SPSA.
The values of a and c control, respectively, the step sizes for parameter updating
and gradient computation. If a is too large, SPSA may overlook a nearby solution and
venture too far away. If a is too small, the algorithm may get stuck locally and never
effectively search the surrounding space. Similarly, a large c may lead parameter
component(s) to hit their bounds (almost) immediately, thus rendering the gradient
approximations invalid. This problem would be magnified if the objective function is
highly non-linear about the current point. On the other hand, a very small value for
c could cause unreliable gradient approximations.
Empirical analyses revealed that suitable values for a and c may be identified by
studying the magnitudes of the gradient approximations and subsequently limiting the
desired updates to certain percentage of the magnitude of each parameter component.
3.8 Summary
In this chapter, typical DTA model calibration variables were described. The process
of data collection was outlined, and the type of data assumed for this research was
discussed. The off-line calibration problem was mathematically formulated, and the
challenging problem dimensions analyzed. The mechanics of three non-linear opti-
mization algorithms with properties suitable to the problem at hand were presented.
An innovative global search approach integrating the Box-Complex and SNOBFIT
algorithms was detailed. The SPSA stochastic approximation algorithm was intro-
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duced as a scalable alternative for large networks. Some theoretical considerations for
the empirical comparison of the computational performances of the two approaches
was outlined, and practical guidelines for the selection of algorithmic terms and con-
stants were outlined. The next two chapters focus on detailed case studies on two
networks: a test case with synthetic data, and a much larger real network from Los
introduction of constraints (in the form of speed observations, in the S(cs) and SD(cs)
estimators) helps fit the speeds better at the expense of the fit to counts. While the
loss of fit to counts is not large, the reduction in RMSEd indicates that better OD
flows have been identified through the information contained in speed data. The
RMSEc value is still below Ref levels, underscoring the overall advantages of the new
approach.
The estimated demand variables in the two SD estimators compare favorably
with the assumed “true” values, as illustrated by the low error in fitting OD flows
as well as the visual comparisons in Figures 4-3 and 4-4. In addition, simultaneous
OD estimation using the traditional assignment matrix formulation1 and Ref supply
parameters resulted in RMSNd = 0.0335, underlining the ability of the proposed
methodology to more accurately capture demand patterns. The travel time coefficient
of the route choice model was estimated as -0.0291/minute and -0.0301/minute for
SD(c) and SD(cs) respectively, representing errors of less than 3% with respect to the
true value of -0.03/minute.
The speed-density parameters estimated for each of the three segment groups are
presented in Tables 4.4 through 4.62. It is verified that the parameter estimates are
stable across all five estimators, and also for all three segment groups.
The calibrated DynaMIT accurately captured the impact of the incident that dis-
abled one of two lanes at the affected location. The estimated segment capacity during
the incident was found to be less than half of the original capacity, consistent with
1Note that the popular sequential OD estimation approach fails in this example, since the traveltime between any origin node and sensor is greater than the departure interval width of five minutes.Thus few, if any, of the vehicles are counted by sensors during their respective departure intervals.
2vmax and vmin are speed estimates measured in miles/hour. kjam and kmin represent densitiesin vehicles/lane-mile. α and β are parameters.
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0 500 1000 1500 2000 2500 3000 3500 4000 45000
500
1000
1500
2000
2500
3000
3500
4000
4500
True OD flows (veh/hour)
Est
imat
ed O
D fl
ows
(veh
/hou
r)
SD(c)
Figure 4-3: Fit to OD Flows (using only counts)
0 500 1000 1500 2000 2500 3000 3500 4000 45000
500
1000
1500
2000
2500
3000
3500
4000
4500
True OD flows (veh/hour)
Est
imat
ed O
D fl
ows
(veh
/hou
r)
SD(cs)
Figure 4-4: Fit to OD Flows (using counts and speeds)
5.5 Synthesis of results and major findings . . . . . . . . . . 160
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5.1 Objectives
Chapter 4 described, through a detailed sensitivity analysis, an evaluation of the
robustness of the calibration methodology developed in this thesis. We concluded
that the new estimators have the ability to replicate demand and supply parameters
underlying several synthetic sensor data sets. A key property of the synthetic network,
however, was the a priori knowledge of the true OD flows and route choice parameter.
Such data are generally not available for real networks. Further, the size of the
problem allowed for extensive experimentation to fine-tune the various algorithms for
rapid convergence.
This chapter documents the second application of the calibration methodology, to
a network in the South Park region of Los Angeles, CA, with the following objectives:
• Demonstrate the feasibility of the estimator on a real network, with unobserved
demand parameters1.
• Demonstrate the scalability of the solution approach on a complex network with
many route choice possibilities.
• Illustrate the advantages of approximation-free simultaneous demand-supply
estimation (without the traditional assignment matrix).
We describe the Los Angeles dataset in some detail, to provide context to this
case study. This includes an analysis of the archived sensor data collected by a real
surveillance system, and a discussion of the extent of variability within the same.
Numerical experiments are outlined, and results comparing the new approach to the
reference case are presented. The tests include a validation exercise illustrating the
benefits of the new off-line methodology in a real-time application.
1Supply parameters remain unobserved, as they were in Chapter 4.
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5.2 The Los Angeles dataset
In this section the network, surveillance data and a log of special events in the study
area are described.
5.2.1 Network description
The Los Angeles network is for the South Park area just south of downtown Los
Angeles. This area includes both the Staples Center and the Los Angeles Convention
Center, and so is heavily affected by special events. Specifically, the area generates a
minimum of 200 events per year ranging from the Democratic National Convention
and the Automobile Show, to NBA Lakers and NHL Kings games. Traffic patterns
vary significantly with different types of events. Recurring commuter traffic along
the Figueroa Corridor, Olympic Boulevard and other one-way streets in the financial
district also pose a challenge to traffic management.
Figure 5-1: The Los Angeles Network
The region is crossed by two major freeways: the Harbor Freeway (I-110) and the
Santa Monica Freeway (I-10). Traffic along the freeways is very heavy throughout the
day, and on weekends. When severe and prolonged traffic congestion develops along
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these freeways, diversions to parallel surface streets frequently occur. The traffic
getting onto the freeways may also be diverted to other ramps connecting to several
major arterials marking the border of the study network (including Hoover Street on
the west, Adams Blvd towards the south, Olympic Blvd due north and Grand Avenue
on the east).
The South Park area has an Advanced Traffic Control System (ATCS) with 109
traffic signals under the control of this new PC-based traffic system. An extensive
video surveillance system and variable message signs are also available to confirm
incidents and provide information to motorists.
The computer representation of the network consisted of 243 nodes connected by
606 directed links. The links were further divided into a total of 740 segments to
capture variations in section geometry and traffic dynamics along the length of each
link.
5.2.2 Surveillance data
The data for this case study was obtained from a set of freeway and arterial loop
detectors that reported time-dependent vehicle counts and detector occupancies for
the month of September 2004. Archived traffic records and location information for
a total of 203 detectors were obtained through two sources. Freeway and ramp data
were extracted from the on-line PeMS (UC Berkeley and Caltrans, 2005) database.
Arterial sensor data were provided by the Los Angeles Department of Transportation
(LADoT). Both sources contained traffic data by lane. Detector occupancies were
converted into density estimates using standard assumptions regarding average vehicle
and detector lengths. Speeds were obtained from counts and densities, using the
fundamental relationship:
q = k v
where q is the flow rate (vehicles/hour); k is the traffic density (vehicles/lane-mile);
v is the space mean speed (miles/hour).
In addition to loop detector data, the surveillance information included a record
142
of incidents that were reported on the network. While the records provided details
such as the incident’s end time, location and general description, they did not contain
any indicators of the start time.
Since the incident start time and duration are key exogenous inputs to the DTA
model, the count data were analyzed to identify abnormalities that could potentially
be ascribed to specific records in the incident log. No such deviations were observed.
While the level of sensor coverage does not preclude major incidents on links without
sensors, the probability of such an event in a relatively small section of the city may
well be low. Minor incidents that occur when flows are below the link capacities
(adjusted to account for the reduction in throughput due to the incident) are not
expected to affect the calibration process, and may be left out of the dataset.
5.2.3 Special events and weather logs
Logs of weather conditions and scheduled special events in and around the study
area were reviewed to identify days expected to have significantly modified travel
demand and/or driver behavior patterns. According to the Weather Underground
website (The Weather Underground, Inc., 2004), there was no precipitation in the Los
Angeles area during the entire month of September 2004 (except for minor showers
on the 14th). Temperatures also remained uniform and high throughout the month.
A list of weekday holidays and special events at the Convention Center was re-
viewed to determine if planned and scheduled events might be a factor in determining
demand patterns. Labor Day counts were found to be markedly different from those
measured on other weekdays in the month. This is to be expected, as the day is
marked by a holiday with a high fraction of shopping trips.
5.2.4 The historical database
A total of one month (September 2004) of freeway and arterial data was analyzed,
to ascertain its sufficiency for the calibration task. Figures 5-2 and 5-3 illustrate
the temporal distribution of sensor counts at two representative counting locations
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on freeway sections, for different days of the week. Figure 5-4 depicts a similar
analysis for a sample arterial sensor. The days were randomly selected from groups
of Mondays, Tuesdays, etc spanning the entire month. It was observed that 5:15-8:00
AM is the most challenging time period from the calibration view-point, as it includes
a sharp (almost linear) build-up of commuter trips over a short duration, and covers
the AM peak period.
Sensor count profiles were compared by time of day and day of the week to help in
the classification of data into day types. Weekdays and weekends displayed markedly
different traffic patterns, with Saturdays and Sundays also significantly different from
each other. Weekdays exhibited similar build-up and dissipation of congestion, with
no clear day-of-the-week effects. The available data was thus classified into three
groups: weekdays, Saturdays and Sundays. The application of the methodology
developed in this thesis is demonstrated for weekdays.
5.3 Application
5.3.1 Reference case
A detailed presentation of the reference case (summarized in Section 2.5) applied to
the Los Angeles dataset is available in Gupta (2005). The work represents the best
current methods applied to off-line DTA system calibration, with the following salient
features:
• Segment capacities (under normal conditions and with incidents) are approxi-
mately determined according to sensor flow data, the number of freeway lanes,
arterial signal timing plans and the recommendations of the Highway Capacity
Manual.
• Segments are classified according to appropriate physical attributes (facility
type, number of lanes, etc.). Speed-density functions for each segment type are
estimated through local regressions between sensor speed and density (occu-
pancy) measurements.
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0 10 20 30 40 50 60 70 80 90 1000
500
1000
1500
2000
2500week 764037
Time of Day (15−min intervals)
Flo
w (
veh/
hour
)
Figure 5-2: Freeway Flows by Day of Week (Sensor ID 764037)
0 10 20 30 40 50 60 70 80 90 1000
500
1000
1500
2000
2500week 718166
Time of Day (15−min intervals)
Flo
w (
veh/
hour
)
Figure 5-3: Freeway Flows by Day of Week (Sensor ID 718166)
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Figure 5-4: Arterial Flows by Day of Week
• The supply parameters (capacities and speed-density functions) are held con-
stant once they are estimated through the steps described above.
• Time-varying OD flows are estimated with a sequential approach using only sen-
sor count data. The restriction on the type of data stems from the use of a linear
assignment matrix mapping between OD flows and link counts. Corresponding
mappings for speed or density measurements are generally intractable.
• The route choice parameter is estimated through a manual line search.
Gupta (2005) describes the application of the above reference methodology to
create a historical database of demand and supply parameters for the entire day
(3:00 AM -midnight).
5.3.2 Network setup and parameters
The time period of 3:00 AM–9:00 AM was selected, so as to include the peak 5:15–
7:00 window and to provide extension into the demand plateau region. Given the low
levels of traffic in the early hours of the day, the period from 3:00-5:15 AM was also
146
used to warm up and load the network. The focus of the evaluation was thus limited
to 5:15-9:00 AM, which was divided into 15-minute time intervals.
A total of nOD = 1129 active OD pairs was identified by Gupta (2005) for the
Los Angeles network, by tracking the non-zero flows over several time intervals of
sequential OD estimation. The flow between every OD pair was estimated for each
time interval in the study period. The set of demand parameters was augmented by
a travel time coefficient used by DynaMIT’s route choice model.
Supply parameters (segment capacities and speed-density function parameters)
were also part of the calibration. Segments were grouped based on physical attributes
such as their position on the network, and the number of lanes in their sections.
Supply parameters were estimated for each group.
5.3.3 Estimators
Five estimators were employed in addition to the reference case. As in Chapter 4, S(c)
and SD(c) correspond to the count-based estimation of supply parameters only, and all
(supply+demand) parameters, respectively2. D1(c) corresponds to the estimation of
demand parameters alone (OD flows and route choice parameters), using Ref supply
parameters. In addition, estimator D2(c) was developed to identify only demand
parameters using the supply parameters from S(c) as given. D2(c) thus corresponds
to sequential demand-supply calibration, as in the reference case. Further, the two D
estimators will help verify the impact of the demand component on the outcome of the
calibration process. Since the OD flows typically outnumber the other parameters,
D would be expected to provide a more significant improvement in fit than S, over
the reference case3. D also includes the effect of simultaneous OD estimation across
intervals, and the direct use of simulator output without linear assignment matrix
approximations. The final estimator, SD(cs), corresponds to simultaneous demand-
supply calibration using both count and speed data.
2Demand parameters from Ref were used while estimating S(c).3This effect was not prominent in the previous case study, due to the relatively small number of
demand parameters.
147
While the combination of S(c) and D2(c) denotes a sequential approach, it should
be noted that the individual demand and supply calibration methods are those de-
veloped in this thesis, and differ markedly from those used in the reference case.
Critically, the new approach eliminates the dependence on an assignment matrix,
thus also providing the flexibility to incorporate any generic traffic measurement into
the calibration framework. Supply calibration is also performed at the network level,
rather than locally at individual sensor locations.
A comparison of D2(c) and SD(c) could provide information about the benefits (if
any) of simultaneous demand-supply calibration over the more traditional sequential
approach.
5.3.4 Measures of performance
The Root Mean Square Normalized (RMSN) error statistic defined in Chapter 4 was
used to document and analyze the performance of the various calibration estimators.
The fit to counts and speeds were computed across all sensors, as well as for freeway
and arterial sensors, to study the accuracy by type of roadway facility. Since true
OD flows, route choice and supply parameters are not available for real networks,
evaluations were limited to the fit to measured data.
Tests of the accuracy of the calibration were augmented with additional validation
analyses. Estimation and prediction tests through a rolling-horizon implementation
of the calibrated DynaMIT system were employed together with a new day of sensor
count measurements, in order to validate the real-time performance obtained as a
result of the above off-line calibration.
5.3.5 Solution algorithm
In Chapter 4, we analyzed the computational effort associated with the Box-SNOBFIT
and SPSA methods as a function of the number of unknowns to be estimated through
calibration. We concluded that the two algorithms estimate comparable parameters,
though SPSA does so at a fraction of the computational cost (measured by running
148
time to convergence) for large-scale problems. The validity of this conclusion was
confirmed by initial tests on the Los Angeles dataset.
The test case involved 7 consecutive time intervals, resulting in K = 4629 vari-
ables. A solution to the problem was attempted with both algorithms, on a dedicated
Pentium 4 processor with 2 GB of physical memory and 750 GB of total hard disk ca-
pacity. The machine ran the Fedora Core 3 Linux operating system. The time taken
for a single function evaluation was approximately 1.5 minutes. While this number
may seem small, a simple calculation of per-iteration effort illustrates the significant
savings provided by SPSA.
A single iteration of SNOBFIT requires K+ 6 = 4635 function evaluations, while
the corresponding number for SPSA is 6 (corresponding to averaging across 3 gra-
†Using Reference supply parameters‡Using S (c) supply parameters
Table 5.1: Fit to Counts: RMSN (15-minute counts)
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The practical usefulness of estimator S extends beyond the above results. Realistic
sensor coverage levels mean that a large fraction of a network’s links are not instru-
mented. Currently, such links (or their constituent segments) are grouped together
with the closest segment type, and a common speed-density relationship is fitted.
Estimator S allows the estimation of a potentially greater number of relationships,
with the closest segment type only providing a priori parameter estimates that may
be revised for better network-wide fit. Further, in applications where only one of the
three basic traffic data (counts, speeds and densities) is available, the reference case
would be faced with insufficient information to derive speed-density relationships.
Estimator S may then be used to update supply parameters transferred from another
location.
D1(c) fits the counts better than S(c), thus highlighting the importance of demand
parameters for calibration. Since demand is the basic and primary driver of traffic
conditions on the network, this result is expected. From an optimization perspective,
the number of unknown OD flows is typically far larger than the supply parameter
set. Consequently, D provides many more degrees of freedom that allow the algorithm
more flexibility in finding a better solution. Interestingly, calibrating demand param-
eters using only count data also results in an improvement in speeds, potentially due
to better density estimates arising from demand patterns that are closer to the true
values.
Demand calibration through D1(c) also provides insight into the limitations of tra-
ditional OD estimation approaches. First, the linearizing assignment matrix trans-
formation of current methods approximates the complex relationship between OD
flows and sensor counts. Second, the often-adopted sequential approach to estimat-
ing OD flows across successive intervals may fail when trip times are much larger
than the width of the estimation interval. A majority of the vehicles departing on
half-hour-long trips, for example, could affect counts in future intervals, and may not
be observed during the fifteen-minute departure interval. The sequential approach
ignores the impact of this lag between a vehicle’s departure time and when it is
actually “measured” on the network. While the assumption may be reasonable on
151
small networks with short trips, it is unrealistic on large and congested networks with
multi-interval trips. The interval width also plays a crucial role, with shorter intervals
accentuating the limitation. Estimator D removes both drawbacks, thus providing
more accurate and efficient OD flow estimates. The improvement in fit for D1(c) over
the reference case highlights this important contribution 4.
D2(c) completes one iteration of sequential demand-supply calibration, in which
the supply parameters obtained from S(c) are used while estimating only OD flows
and route choice model parameters through the D estimator. The results show better
fit than either S(c) or D1(c), indicating the benefits of joint model calibration. Fur-
ther iterations between the demand and supply estimators may be performed until
convergence criteria are satisfied. However, it must be remembered that each itera-
tion consists of two complex, large-scale optimization problems. Further, the rate of
convergence of this iterative method is difficult to establish, and the expected number
of iterations is consequently unavailable.
SD(c) represents the simultaneous equivalent of D2(c), with both demand and
supply parameters estimated together. This estimator thus does not involve demand-
supply iterations, and terminates after the solution of a single optimization problem.
This approach is preferable, as it provides both added efficiency (through simulta-
neous calibration) and rapid convergence. SD(c) improves upon D2(c), though the
reduction in RMSN is relatively small. It should however be preferred in practice,
owing to the advantages outlined earlier.
The final estimator, SD(cs), extends the SD(c) case to match speed observa-
tions in addition to counts. As in the previous case study, the introduction of speed
measurement equations results in a marginal loss of fit to counts. This must be an-
ticipated, as the objective function being minimized now includes additional terms.
From a practical perspective, the speed information plays the role of selecting the
right set of demand parameters among many that may capture the counts accurately.
In Park et al. (2005), for example, traditional count-based OD estimation reliably
4D1(c) was estimated with reference supply parameters. The improvement in fit over the referencecase therefore captures the differences in demand estimation between the two approaches.
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captured temporal count profiles, yet failed to replicate path travel times measured
using probe vehicles. A calibration methodology that can significantly improve the
model’s ability to explain traffic dynamics, with minimum impact to the fit to counts,
would thus be invaluable in travel time reliability studies and route guidance applica-
tions. The fit to speed data improves significantly from SD(c) to SD(cs), underscoring
a key contribution of the proposed calibration methodology arising from the ability
to include general traffic data.
Figure 5-5 compares the fitted counts from Ref and SD(c), against the actual
counts for all sensors and time intervals. The fit to sensor count data was also analyzed
by time of day, to ensure that the calibration methodology effectively tracked the
profiles of observed counts across the network. Figure 5-6 shows cumulative counts
(summed across all sensors). Visual inspection illustrates the accuracy of SD(c), and
the improvement over Ref.
Some sample plots of cumulative counts at individual sensor locations are pre-
sented in Figures 5-7 to 5-12, which further indicate the accuracy of the calibration
approach developed in this thesis. Indeed, the SD(c) case results in a more accurate
fit to counts when compared with the reference case. The six sensors shown here were
selected at random to provide a spatial spread across the entire network.
The fit to count data across the entire network is illustrated in Figure 5-13.
RMSNc values for a sample of sensors distributed spatially on both freeways and
arterials are provided, as further proof of calibration accuracy.
5.4.2 Validation results
The results in the previous section illustrated the ability of the methodology to better
fit observed data in an off-line scenario. Additional validation tests were performed
to evaluate the ability of the calibrated system to provide real-time traffic estimations
and predictions of higher quality compared to the reference case. For this purpose, the
DynaMIT-R DTA system was run in a rolling horizon, with sensor count data from
a new day (not used for calibration) used to simulate the real-time data collection